Environmental Engineering Reference
In-Depth Information
Table 2.10
First de
rivatives of the selected four copulas
Copula First derivative of C (u
1
, u
2
;
θ
) with respect to u
2
, M (u
1
, u
2
;
θ
)
Gaussian
Φ
1
()
u
1
()
u
Φ
−
−
−
θ
θ
Φ
−
1
2
1
2
Plackett
1
2
1
(
1
)
u
(
1
)
u
+− −+
+− +
θ
θ
2
1
−
21
{[
(
1
)(
uu
)]
2
4
uu
(
1
)}
(/ )
12
θ
−
θ θ
−
1
2
12
Frank
e
u
(
e
u
1
)
−
θ
−
θ
−
2
1
(
e
−
θ
1
)
(
e
−
θ
u
1
)(
e
−
θ
u
1
)
−+ −
−
1
2
No.16
1
2
1
2
=+−− +−
11
1
θ
++
−
1
1
SS
(
2
4
)
,
Suu
1
+
θ
θ
1
2
uu
u
2
1
2
derivatives of the four copulas considered. When the copula parameters θ are known, the
probabilities of failure for the retaining wall can be efficiently evaluated using
Equation 2.41
.
2.5.3 nominal factor of safety for retaining wall stability
It is clear that a mean factor of safety computed by substituting mean values for the ran-
dom variables in
Equation 2.31
c
annot account for the COVs of shear strength parameters
and the correlation between cohesion and friction angle. To take into consideration these
statistics approximately, nominal factors of safety involving cautious estimates of the shear
strength parameters are introduced. Following the Eurocode 7 practice (Orr 2000), a 5%
fractile value of the factors of safety shown in
Equation 2.31
is defined as the nominal factor
of safety. In this chapter, the nominal factor of safety,
FS
n
, is obtained from simulations as
illustrated below using the Plackett copula (Tang et al. 2013a). The algorithm for simulating
the nominal factor of safety consists of the following three steps:
1. Simulate two correlated standard uniform vectors
u
m
×2
= [
U
1
,
U
2
] belonging to the
Plackett copula using the algorithms in Section 2.4.1. Note that a sample size of
m
= 10
6
is adopted for the simulation.
2. Let
X
m
F
2
−
are the inverse CDFs
of
c
and ϕ, respectively. In this example,
c
and ϕ are Lognormal variables.
=
(
c
, φ
)
=
(
F
−
1
(
UF U
),
−
1
( )
in which
F
1
−
(. and
()
×
2
1
12
2
the retaining wall is obtained. The 5% fractile value of the simulated factors of safety
is obtained using the MATLAB function
quantile(FS,0.05)
, which is taken as
FS
n
.
It is evident from the above simulation procedures that
FS
n
is a function of deterministic
parameters (e.g., geometrical parameters
a
and
b
) and statistical parameters (e.g., COV and
ρ of shear strength parameters). In the parametric studies presented below, the variation
of
p
f
with various deterministic and statistical parameters is studied by plotting against
FS
n
rather than the mean factor of safety. There are two reasons for this more complicated
choice. First, a single horizontal axis based on
FS
n
can be applied in all parameter studies
(deterministic and statistical parameters), thus providing a unified and concise presentation
of the results. On the other hand, the mean factor of safety is just a function of deterministic
parameters. If
p
f
is plotted against the mean factor of safety, only the variations of
p
f
with
Search WWH ::
Custom Search